Solve
\[(x^3 + 3x^2 \sqrt{2} + 6x + 2 \sqrt{2}) + (x + \sqrt{2}) = 0.\]Enter all the solutions, separated by commas.
Explanation: We can write the equation as
\[(x + \sqrt{2})^3 + (x + \sqrt{2}) = 0.\]Then
\[(x + \sqrt{2})[(x + \sqrt{2})^2 + 1] = 0,\]so $x = -\sqrt{2}$ or $(x + \sqrt{2})^2 = -1.$  For the latter equation,
\[x + \sqrt{2} = \pm i,\]so $x = -\sqrt{2} \pm i.$

Thus, the solutions are $\boxed{-\sqrt{2}, -\sqrt{2} + i, -\sqrt{2} - i}.$